ON THE OSCILLATION OF CERTAIN THIRD-ORDER
DIFFERENCE EQUATIONS
RAVI P. AGARWAL, SAID R. GRACE, AND DONAL O’REGAN
Received 28 August 2004
We establish some new criteria for the oscillation of third-order difference equations of
the form ∆((1/a2 (n))(∆(1/a1 (n))(∆x(n))α1 )α2 ) + δq(n) f (x[g(n)]) = 0, where ∆ is the forward difference operator defined by ∆x(n) = x(n + 1) − x(n).
1. Introduction
In this paper, we are concerned with the oscillatory behavior of the third-order difference
equation
L3 x(n) + δq(n) f x g(n)
= 0,
(1.1;δ)
where δ = ±1, n ∈ N = {0,1,2,...},
L0 x(n) = x(n),
L2 x(n) =
L1 x(n) =
α
1 ∆L1 x(n) 2 ,
a2 (n)
α
1 ∆L0 x(n) 1 ,
a1 (n)
L3 x(n) = ∆L2 x(n).
(1.2)
In what follows, we will assume that
(i) {ai (n)}, i = 1,2, and {q(n)} are positive sequences and
∞
ai (n)
1/αi
= ∞,
i = 1,2;
(1.3)
(ii) {g(n)} is a nondecreasing sequence, and limn→∞ g(n) = ∞;
(iii) f ∈ Ꮿ(R, R), x f (x) > 0, and f (x) ≥ 0 for x = 0;
(iv) αi , i = 1,2, are quotients of positive odd integers.
The domain Ᏸ(L3 ) of L3 is defined to be the set of all sequences {x(n)}, n ≥ n0 ≥ 0
such that {L j x(n)}, 0 ≤ j ≤ 3 exist for n ≥ n0 .
A nontrivial solution {x(n)} of (1.1;δ) is called nonoscillatory if it is either eventually
positive or eventually negative and it is oscillatory otherwise. An equation (1.1;δ) is called
oscillatory if all its nontrivial solutions are oscillatory.
Copyright © 2005 Hindawi Publishing Corporation
Advances in Difference Equations 2005:3 (2005) 345–367
DOI: 10.1155/ADE.2005.345
346
On the oscillation of certain third-order difference equations
The oscillatory behavior of second-order half-linear difference equations of the form
∆
α
1 ∆x(n) 1 + δq(n) f x g(n) = 0,
a1 (n)
(1.4;δ)
where δ, a1 , q, g, f , and α1 are as in (1.1;δ) and/or related equations has been the subject of intensive study in the last decade. For typical results regarding (1.4;δ), we refer
the reader to the monographs [1, 2, 4, 8, 12], the papers [3, 6, 11, 15], and the references cited therein. However, compared to second-order difference equations of type
(1.4;δ), the study of higher-order equations, and in particular third-order equations of
type (1.1;δ) has received considerably less attention (see [9, 10, 14]). In fact, not much
has been established for equations with deviating arguments. The purpose of this paper
is to present a systematic study for the behavioral properties of solutions of (1.1;δ), and
therefore, establish criteria for the oscillation of (1.1;δ).
2. Properties of solutions of equation (1.1;1)
We will say that {x(n)} is of type B0 if
x(n) > 0,
L1 x(n) < 0,
L2 x(n) > 0,
L3 x(n) ≤ 0
eventually,
(2.1)
L1 x(n) > 0,
L2 x(n) > 0,
L3 x(n) ≤ 0
eventually.
(2.2)
it is of type B2 if
x(n) > 0,
Clearly, any positive solution of (1.1;1) is either of type B0 or B2 . In what follows, we
will present some criteria for the nonexistence of solutions of type B0 for (1.1;1).
Theorem 2.1. Let conditions (i)–(iv) hold, g(n) < n for n ≥ n0 ≥ 0, and
− f (−xy) ≥ f (xy) ≥ f (x) f (y)
for xy > 0.
(2.3)
Moreover, assume that there exists a nondecreasing sequence {ξ(n)} such that g(n) < ξ(n)
< n for n ≥ n0 . If all bounded solutions of the second-order half-linear difference equation


1/α
α
1 ∆y(n) 2 − q(n) f 
a1 1 (k) f y 1/α1 ξ(n) = 0
∆
a2 (n)
k=g(n)
ξ(n)
(2.4)
are oscillatory, then (1.1;1) has no solution of type B0 .
Proof. Let {x(n)} be a solution of (1.1;1) of type B0 . There exists n0 ∈ N so large that
(2.1) holds for all n ≥ n0 . For t ≥ s ≥ n0 , we have
x(s) = x(t + 1) −
t
j =s

1
a1/α
1 ( j)
1
1
a1/α
1 ( j)
∆x( j) ≥ 
t
j =s

1
1
 − L1/α
a1/α
1 ( j)
1 x(t) .
(2.5)
Ravi P. Agarwal et al. 347
Replacing s and t by g(n) and ξ(n) respectively in (2.5), we have

x g(n) ≥ 
ξ(n)

1

a1/α
1 ( j)
1/α1
− L1
x ξ(n)
(2.6)
j =g(n)
for n ≥ n1 ∈ N for some n1 ≥ n0 . Now using (2.3) and (2.6) in (1.1;1) and letting y(n) =
−L1 x(n) > 0 for n ≥ n1 , we easily find


1/α
α
1 ∆y(n) 2 − q(n) f 
a1 1 ( j) f y 1/α1 ξ(n) ≥ 0 for n ≥ n1 .
∆
a2 (n)
j =g(n)
ξ(n)
(2.7)
A special case of [16, Lemma 2.4] guarantees that (2.4) has a positive solution, a contra
diction. This completes the proof.
Theorem 2.2. Let conditions (i)–(iv) and (2.3) hold, and assume that there exists a nondecreasing sequence {ξ(n)} such that g(n) < ξ(n) < n for n ≥ n0 . Then, (1.1;1) has no solution
of type B0 if either one of the following conditions holds:
(S1 )
f u1/(α1 α2 )
≥ 1 for u = 0,
u



(2.8)
 
1/α1 

ξ(n)



1/α1
1/α2



limsup
q(k)
f
a
(
j)
f
a
(i)
> 1,


1
2



n→∞ k=ξ(n) 
j =g(k)
i=ξ(k)
n
−1

ξ(k)
(2.9)
(S2 )
f
u
u1/(α1 α2 )
−→ 0
as u −→ 0,
(2.10)
 
1/α1 

ξ(n)


1/α1
1/α2

 f 
 
limsup
q(k)
f
a
(
j)
a
(i)
> 0.

1
2



n→∞ k=ξ(n) 
j =g(k)
i=ξ(k)
n
−1




ξ(k)
(2.11)
Proof. Let {x(n)} be a solution of (1.1;1) of type B0 . Proceeding as in the proof of
Theorem 2.1 to obtain the inequality (2.7), it is easy to check that y(n) > 0 and ∆y(n) < 0
for n ≥ n1 . Let n2 > n1 be such that inf n≥n2 ξ(n) > n1 . Now
y(σ) = y(τ + 1) −

≥
τ
2
a1/α
2 ( j)
j =σ
τ
j =σ

2

a1/α
2 ( j)
α
1 ∆y( j) 2
a2 ( j)
α2
1 − ∆y(τ)
a2 (τ)
1/α2
(2.12)
1/α2
for τ ≥ σ ≥ n2 .
348
On the oscillation of certain third-order difference equations
Replacing σ and τ by ξ(k) and ξ(n) respectively in (2.12), we have

y ξ(k) ≥ 

ξ(n)
2

a1/α
2 ( j)
j =ξ(k)
α
1
− ∆y ξ(n) 2
a2 ξ(n)
1/α2
for n ≥ k ≥ n2 .
(2.13)
Summing (2.7) from ξ(n) to (n − 1) and letting Y (n) = (−∆y(n))α2 /a2 (n) for n ≥ n2 , we
get

n
−1
Y ξ(n) ≥ Y (n) +
q(k) f 
1

a1/α
1 ( j)
1/α1 
2
 1/α2 ξ(n)  
a1/α

2 (i) Y

ξ(n)

× f 

j =g(k)
k=ξ(n)

ξ(k)
(2.14)
for n ≥ n2 .
i=ξ(k)
Using condition (2.3) in (2.14), we have
Y ξ(n) ≥ f Y 1/(α1 α2 ) ξ(n)


×
n
−1

q(k) f 
ξ(k)
 
1/α1 
ξ(n)

1
2
 
 
a1/α
a1/α
 ,
1 ( j) f 
2 (i)
j =g(k)
k=ξ(n)
n ≥ n2 .
i=ξ(k)
(2.15)
Using (2.8) in (2.15) we have
1≥

n
−1
q(k) f 
k=ξ(n)
ξ(k)
 
1/α1 
ξ(n)
 1/α2  
f
a2 (i)
.
1

a1/α
1 ( j)
j =g(k)
(2.16)
i=ξ(k)
Taking limsup of both sides of the above inequality as n → ∞, we obtain a contradiction
to condition (2.9).
Next, using (2.10) in (2.15) and taking limsup of the resulting inequality, we obtain a
contradiction to condition (2.11). This completes the proof.
Theorem 2.3. Let the hypotheses of Theorem 2.2 hold. Then, (1.1;1) has no solutions of type
B0 if one of the following conditions holds:
(O1 )
f 1/α2 u1/α1
≥ 1 for u = 0,
u
limsup
n→∞
n
−1
k=ξ(n)

2

a1/α
2 (k)
n
−1
j =k

q( j) f 
ξ( j)
i=g( j)
(2.17)
1/α2
1

a1/α
1 (i)
> 1,
(2.18)
Ravi P. Agarwal et al. 349
(O2 )
u
−→ 0 as u −→ 0,
f 1/α2 u1/α1

n
−1
n
−1
2

a1/α
(k)
q( j) f
limsup
2
n→∞ k=ξ(n)
j =k


ξ( j)
(2.19)
1/α2
1

a1/α
1 (i)
> 0.
(2.20)
i=g( j)
Proof. Let {x(n)} be a solution of (1.1;1) of type B0 . As in the proof of Theorem 2.1, we
obtain the inequality (2.7) for n ≥ n1 . Also, we see that y(n) > 0 and ∆y(n) < 0 for n ≥ n1 .
Next, we let n2 ≥ n1 be as in the proof of Theorem 2.2, and summing inequality (2.7)
from s ≥ n2 to (n − 1), we have

n −1

1/α
α2
α2 1 1 q(k) f 
a1 1 ( j) f y 1/α1 ξ(k) ,
− ∆y(s) ≥
− ∆y(n) +
a2 (s)
a2 (n)
k =s
j =g(k)
(2.21)
which implies

1/α
−∆y(s) ≥ a2 2 (s) 
n
−1

q(k) f 
k =s
ξ(k)

1

a1/α
1 ( j)
ξ(k)
f y
1/α1
1/α2
ξ(k)  .
(2.22)
j =g(k)
Now,
y(v) = y(n) +
n
−1
s =v
−1
n
− ∆y(s) ≥
− ∆y(s)
for n − 1 ≥ s ≥ n2 .
(2.23)
s =v
Substituting (2.23) in (2.22) and setting v = ξ(n), we have
y ξ(n)

n
−1
n
−1
2
 q(k) f
a1/α
≥
2 (s)
k =s
s=ξ(n)
≥ f 1/α2 y 1/α1 ξ(n)
−1
n
s=ξ(n)


ξ(k)

1

a1/α
1 ( j)
f y 1/α1
1/α2
ξ(k) 
j =g(k)

2

a1/α
2 (s)
n
−1
k =s

q(k) f 
ξ(k)
(2.24)
1/α2
1

a1/α
1 ( j)
.
j =g(k)
The rest of the proof is similar to that of Theorem 2.2 and hence is omitted.
350
On the oscillation of certain third-order difference equations
Theorem 2.4. Let conditions (i)–(iv), (2.3) hold, g(n) = n − τ, where τ is a positive integer
and assume that there exist two positive integers such that τ > τ > τ̃. If the first-order delay
equation

∆y(n) + q(n) f 
n
−τ
 
1/α1 
n
−τ̃
1
2
 
 
a1/α
a1/α
 f y 1/(α1 α2 ) [n − τ̃] = 0
1 ( j) f 
2 (i)
j =n−τ
i=n−τ
(2.25)
is oscillatory, then (1.1;1) has no solution of type B0 .
Proof. Let {x(n)} be a solution of (1.1;1) of type B0 . As in the proof of Theorem 2.1, we
obtain (2.6) for n ≥ n1 , which takes the form

x[n − τ] ≥ 
n
−τ

1

a1/α
1 ( j)
1/α1
− L1
x[n − τ]
for n ≥ n1 .
(2.26)
for n ≥ n2 ≥ n1 .
(2.27)
j =n −τ
Similarly, we find

−L1 x[n − τ] ≥ 
n
−τ̃

2
2
 L1/α
a1/α
2 (i)
2 x[n − τ̃]
i=n−τ
Combining (2.26) with (2.27) we have

x[n − τ] ≥ 
n
−τ

n
−τ̃
1

a1/α
1 ( j)
j =n−τ
1/α1
2

a1/α
2 (i)
1 α2 )
L1/(α
x[n − τ̃]
2
for n ≥ n3 ≥ n2 .
i =n −τ
(2.28)
Using (2.3) and (2.28) in (1.1;1) and setting Z(n) = L2 x(n), we have

∆Z(n) + q(n) f 
×f Z
n
−τ
j =n−τ
1/(α1 α2 )
 
1/α1 
n
−τ̃
1
2
 
 
a1/α
a1/α

1 ( j) f 
2 (i)
i =n −τ
(2.29)
[n − τ̃] ≤ 0 for n ≥ n3 .
By a known result in [2, 12], we see that (2.25) has a positive solution which is a contra
diction. This completes the proof.
As an application of Theorem 2.4, we have the following result.
Ravi P. Agarwal et al. 351
Corollary 2.5. Let conditions (i)–(iv), (2.3) hold, g(n) = n − τ, τ is a positive integer and
let there exist two positive integers τ, τ̃ such that τ > τ > τ̃. Then, (1.1;1) has no solution of
type B0 if either one of the following conditions holds:
(I1 ) in addition to (2.8),
liminf
n→∞

n
−1
q(k) f 
k =n −τ
 
1/α1 
k−τ̃
τ̃ τ̃+1
 1/α2  
f
a2 (i)
,
>
k
−τ
1

a1/α
1 ( j)
j =k−τ
1 + τ̃
i =k −τ
(2.30)
(I2 )
∞
k =n 0

q(k) f 
±0
k
−τ
du
< ∞,
f u1/(α1 α2 )
(2.31)
 
1/α1 
k
−τ̃

1
2


 
a1/α
a1/α
 = ∞.
1 ( j) f 
2 (i)
j =k−τ
(2.32)
i=k−τ
Next, we will present some criteria for the nonexistence of solutions of type B2 of
(1.1;1).
Theorem 2.6. Let conditions (i)–(iv) and (2.3) hold. If
∞

q( j) f 
g( j)−1

1

a1/α
1 (i) = ∞,
(2.33)
i=n0
then (1.1;1) has no solution of type B2 .
Proof. Let {x(n)} be a solution of (1.1;1). There exists an integer n0 ∈ N so large that
(2.2) holds for n ≥ n0 . From (2.2), there exist a constant c > 0 and an integer n1 ≥ n0 such
that
α
1 ∆L0 x(n) 1 = L1 x(n) ≥ c,
a1 (n)
(2.34)
or
∆x(n) ≥ ca1 (n)
1/α1
for n ≥ n1 .
(2.35)
Summing (2.35) from n1 to g(n) − 1(≥ n1 ) we obtain
g(n)−1
x g(n) ≥ c1/α1
1
a1/α
1 ( j).
(2.36)
j =n1
Using (2.3) and (2.36) in (1.1;1) we have
−L3 x(n) = q(n) f x[g(n)]


g(n)−1
1/α
1/α ≥ q(n) f c 1 f 
a1 1 ( j)
j =n1
for n ≥ n2 ≥ n1 .
(2.37)
352
On the oscillation of certain third-order difference equations
Summing (2.37) from n2 to n − 1(> n2 ) we obtain
∞ > L2 x(n2 ) ≥ −L2 x(n) + L2 x(n2 )


g(k)−1
−1
1/α
1/α n
1
1
≥f c
q(k) f 
a1 ( j) −→ ∞
as n −→ ∞,
(2.38)
j =n1
k =n 2
a contradiction. This completes the proof.
Theorem 2.7. Let conditions (i)–(iv) and (2.3) hold, and g(n) = n − τ, n ≥ n0 ≥ 0, where
τ is a positive integer. If the first-order delay equation


∆y(n) + q(n) f 
n−
τ −1

a1 (k)
k
−1
1/α1 
2
 
a1/α
 f y 1/(α1 α2 ) [n − τ] = 0
2 ( j)
(2.39)
j =n0
k =n 0
is oscillatory, then (1.1;1) has no solution of type B2 .
Proof. Let {x(n)} be a solution of (1.1;1) of type B2 . There exists an integer n0 ≥ 0 so large
that (2.2) holds for n ≥ n0 . Now,
L1 x(n) = L1 x(n0 ) +
n
−1
∆L1 x( j)
j =n0
= L1 x(n0 ) +
n
−1
−1/α2
2
a1/α
( j)∆L1 x( j)
2 ( j) a2
j =n0
= L1 x(n0 ) +
n
−1
(2.40)
1/α2
2
a1/α
2 ( j)L2 x( j)
j =n0
1/α2
≥ L2
x(n)
n
−1
2
a1/α
2 ( j) for n ≥ n1 ,
j =n 0
or
n −1
1/α
α
1 2
∆x(n) 1 ≥ L1/α
a2 2 ( j).
2 x(n)
a1 (n)
j =n 0
(2.41)
Thus,

∆x(n) ≥ a1 (n)
n
−1
1/α1
2

a1/α
2 ( j)
1 α2 )
L1/(α
x(n) for n ≥ n0 .
2
(2.42)
j =n0
Summing (2.42) from n0 to g(n) − 1 > n0 , we have


1/α1 
k
−1

1/(α α ) 2
a1 (k)
 
x g(n) ≥ 
a1/α
 L2 1 2 x g(n)
2 ( j)
g(n)−1
k =n 0
j =n 0
for n ≥ n1 ≥ n0 .
(2.43)
Ravi P. Agarwal et al. 353
Using (2.3), (2.43), g(n) = n − τ, and letting y(n) = L2 x(n), n ≥ n1 , we obtain


1/α1 
k −τ −1
k
−1
 

1/α2
a1 (k)
a2 ( j)  f y 1/(α1 α2 ) [n − τ] ≤ 0.
∆y(n) + q(n) f 
(2.44)
j =n0
k =n 0
The rest of the proof is similar to that of Theorem 2.4 and hence is omitted.
Theorem 2.8. Let conditions (i)–(iv) and (2.3) hold and g(n) > n + 1 for n ≥ n0 ∈ N. If the
half-linear difference equation


g(n)−1
1/α
α2
1 
∆y(n)
+ q(n) f
a1 1 ( j) f y 1/α1 (n) = 0
∆
a2 (n)
j =n
(2.45)
is oscillatory, then (1.1;1) has no solution of type B2 .
Proof. Let {x(n)} be a solution of (1.1;1) of type B2 . Then there exists an n0 ∈ N sufficiently large so that (2.2) holds for n ≥ n0 . Now, for m ≥ s ≥ n0 we get
x(m) − x(s) =
m
−1
1/α1
1
a1/α
1 ( j)L1 x( j),
(2.46)
j =s
or


m
−1
x(m) ≥ 
1
 1/α1
a1/α
1 ( j) L1 x(s).
(2.47)
j =s
Replacing m and s in (2.47) by g(n) and n, respectively, we have

x g(n) ≥ 
g(n)−1

1
 1/α1
a1/α
1 ( j) L1 x(n)
for g(n) ≥ n + 1 ≥ n1 ≥ n0 .
(2.48)
j =n
Using (2.3) and (2.48) in (1.1;1) and letting Z(n) = L1 x(n) for n ≥ n1 , we obtain


g(n)−1
1/α
α
1 ∆
∆Z(n) 2 + q(n) f 
a1 1 ( j) f Z 1/α1 (n) ≤ 0 for n ≥ n1 .
a2 (n)
j =n
(2.49)
By [16, Lemma 2.3], we see that (2.45) has a positive solution, a contradiction. This com
pletes the proof.
Remark 2.9. We note that a corollary similar to Corollary 2.5 can be deduced from
Theorem 2.7. Here, we omit the details.
Remark 2.10. We note that the conclusion of Theorems 2.1–2.4 can be replaced by “all
bounded solutions of (1.1;1) are oscillatory.”
Next, we will combine our earlier results to obtain some sufficient conditions for the
oscillation of (1.1;1).
354
On the oscillation of certain third-order difference equations
Theorem 2.11. Let conditions (i)–(iv) and (2.3) hold, g(n) < n for n ≥ n0 ∈ N. Moreover,
assume that there exists a nondecreasing sequence {ξ(n)} such that g(n) < ξ(n) < n for n ≥
n0 . If either conditions (S1 ) or (S2 ) of Theorem 2.2 and condition (2.33) hold, the equation
(1.1;1) is oscillatory.
Proof. Let {x(n)} be a nonoscillatory solution of (1.1;1), say, x(n) > 0 for n ≥ n0 ∈ N.
Then, {x(n)} is either of type B0 or B2 . By Theorem 2.2, {x(n)} is not of type B0 and by
Theorem 2.6, {x(n)} is not of type B2 . This completes the proof.
Theorem 2.12. Let conditions (i)–(iv), (2.3) hold, g(n) = n − τ, n ≥ n0 ∈ N, where τ is
a positive integer. Moreover, assume that there exist two positive integers τ and τ̃ such that
τ > τ > τ̃. If both first-order delay equations (2.25) and (2.39) are oscillatory, then (1.1;1) is
oscillatory.
Proof. The proof follows from Theorems 2.4 and 2.7.
Next, we will apply Theorems 2.11 and 2.12 to a special case of (1.1;1), namely, the
equation
α
1 1
∆
∆x(n) 1
∆
a2 (n) a1 (n)
α2 + q(n)xα g(n) = 0,
(2.50)
where α is the ratio of positive odd integers.
Corollary 2.13. Let conditions (i)–(iv) hold, g(n) < n for n ≥ n0 ∈ N, and assume that
there exists a nondecreasing sequence {ξ(n)} such that g(n) < ξ(n) < n for n ≥ n0 . Equation
(2.50) is oscillatory if either one of the following conditions holds:
(A1 ) α = α1 α2 ,
∞
j =n0 ≥0
limsup
n→∞
n
−1
j =ξ(n)

g( j)−1
q( j) 
α
1

a1/α
1 (i)
i =n 0

ξ( j)
q( j) 
α 
1

a1/α
1 (i)
i=g( j)
= ∞,
ξ(n)

(2.51)
α2
2

a1/α
2 (i)
> 1,
(2.52)
> 0.
(2.53)
i=ξ( j)
(A2 ) α < α1 α2 and condition (2.51) hold, and
limsup
n→∞
n
−1
j =ξ(n)

q( j) 
ξ( j)
i=g( j)
α 
1

a1/α
1 (i)

ξ(n)
α2
2

a1/α
2 (i)
i=ξ( j)
Corollary 2.14. Let conditions (i)–(iv) hold, g(n) = n − τ, n ≥ n0 ∈ N, where τ is a positive integer, and assume that there exist two positive integers τ, τ̃ such that τ > τ > τ̃. If
Ravi P. Agarwal et al. 355
the first-order delay equations

α 
α2
n
−τ̃
1/α
1/α
1
2
a1 ( j) 
a2 (i) Z α/(α1 α2 ) [n − τ̃] = 0,
∆y(n) + q(n) 
j =n−τ
i=n−τ


1/α1 α
j −1
n−
τ −1

2
a1 ( j)
 
a1/α
∆Z(n) + q(n) 
 Z α/(α1 α2 ) [n − τ] = 0
2 (i)
j =n 0
i=n0
n
−τ
(2.54)
(2.55)
are oscillatory, then (2.50) is oscillatory.
For the mixed difference equations of the form
L3 x(t) + q1 (t) f1 x g1 (n) + q2 (n) f2 x g2 (n)
= 0,
(2.56)
where L3 is defined as in (1.1;1), {ai (n)}, i = 1,2 are as in (i) satisfying (1.3), α1 and α2
are as in (iv), {qi (n)}, i = 1,2 are positive sequences, {gi (n)}, i = 1,2 are nondecreasing
sequences with limn→∞ gi (n) = ∞, i = 1,2, fi ∈ Ꮿ(R, R), x fi (x) > 0 and fi (x) ≥ 0 for x = 0
and i = 1,2. Also, f1 , f2 satisfy condition (2.3) by replacing f by f1 and/or f2 .
Now, we combine Theorems 2.1 and 2.8 and obtain the following interesting result.
Theorem 2.15. Let the above hypotheses hold for (2.56), g1 (n) < n and g2 (n) > n + 1 for
n ≥ n0 ∈ N and assume that there exists a nondecreasing sequence {ξ(n)} such that g1 (n) <
ξ(n) < n for n ≥ n0 . If all bounded solutions of the equation


1/α
α
1 ∆y(n) 2 − q1 (n) f1 
a1 1 (k) f1 y 1/α1 ξ(n) = 0
∆
a2 (n)
k=g (n)
ξ(n)
(2.57)
1
are oscillatory and all solutions of the equation


g(n)−1
1/α
α
1 ∆Z(n) 2 + q2 (n) f2 
a1 1 ( j) f2 Z 1/α1 (n) = 0
∆
a2 (n)
j =n
(2.58)
are oscillatory, then (2.56) is oscillatory.
3. Properties of solutions of equation (1.1;-1)
We will say that {x(n)} is of type B1 if
x(n) > 0,
L1 x(n) > 0,
L2 x(n) < 0,
L3 x(n) ≥ 0
eventually,
(3.1)
it is of type B3 if
x(n) > 0,
Li x(n) > 0,
i = 1,2,
L3 x(n) ≥ 0 eventually.
(3.2)
Clearly, any positive solution of (1.1;-1) is either of type B1 or B3 . In what follows, we
will give some criteria for the nonexistence of solutions of type B1 for (1.1;-1).
356
On the oscillation of certain third-order difference equations
Theorem 3.1. Assume that conditions (i)–(iv) hold. If
∞
q( j) = ∞,
(3.3)
then (1.1;-1) has no solution of type B1 .
Proof. Let {x(n)} be a solution of (1.1;-1) of type B1 . Then there exists an n0 ∈ N sufficiently large so that (3.1) holds for n ≥ n0 . Next, there exist an integer n1 ≥ n0 and a
constant c > 0 such that
for n ≥ n1 .
x g(n) ≥ c
(3.4)
Summing (1.1;-1) from n1 to n − 1 ≥ n1 and using (3.4), we have
L2 x(n) − L2 x(n1 ) =
n
−1
q( j) f x g( j) ,
(3.5)
j =n 1
or
∞ > −L2 x(n1 ) ≥ f (c)
n
−1
q( j) −→ ∞
as n −→ ∞,
(3.6)
j =n1
a contradiction. This completes the proof.
Theorem 3.2. Let conditions (i)–(iv) and (2.3) hold and g(n) < n for n ≥ n0 ∈ N. If all
bounded solutions of the half-linear equation


g(n)−1
1/α
α
1 ∆y(n) 2 − q(n) f 
a1 1 ( j) f y 1/α1 g(n) = 0
∆
a2 (n)
j =n 0
(3.7)
are oscillatory, then (1.1;-1) has no solutions of type B1 .
Proof. Let {x(n)} be a solution of (1.1;-1) of type B1 . There exists an n0 ∈ N such that
(3.1) holds for n ≥ n0 . Now
n
−1
x(n) − x(n0 ) =
∆x( j) =
j =n0
n
−1
1/α1
1
a1/α
1 ( j)L1 x( j).
(3.8)
j =n 0
Thus,

x(n) ≥ 
n
−1

1
 1/α1
a1/α
1 ( j) L1 x(n)
for n ≥ n0 .
(3.9)
j =n 0
There exists an n1 ≥ n0 such that

x g(n) ≥ 
g(n)−1
j =n 0

1
 1/α1
a1/α
1 ( j) L1 x g(n)
for n ≥ n1 .
(3.10)
Ravi P. Agarwal et al. 357
Using (2.3) and (3.10) in (1.1;-1) and letting y(n) = L1 x(n) for n ≥ n1 , we have


g(n)−1
1/α
α
1 ∆
∆y(n) 2 ≥ q(n) f 
a1 1 ( j) f y 1/α1 g(n)
a2 (n)
j =n 0
for n ≥ n1 .
The rest of the proof is similar to that of Theorem 2.1 and hence is omitted.
(3.11)
Next, we state the following criteria which are similar to Theorems 2.2, 2.3, and 2.4.
Here, we omit the proofs.
Theorem 3.3. Let conditions (i)–(iv) and (2.3) hold, and g(n) < n for n ≥ n0 ∈ N. Then,
(1.1;-1) has no solution of type B1 if either one of the following conditions holds:
(C1 ) condition (2.8) holds, and
 
1/α1 

g(n)



1/α1
1/α2




limsup
q(k)
f
a
(
j)
f
a
(i)
> 1,


1
2



n→∞ k=g(n) 
j =n0 ≥0
i=g(k)
n
−1




g(k)−1
(3.12)
(C2 ) condition (2.10) holds, and
 
1/α1 

g(n)



1/α1
1/α2



limsup
q(k)
f
a
(
j)
f
a
(i)
> 0.


1
2



n→∞ k=g(n) 
j =n0 ≥0
i=g(k)
n
−1




g(k)−1
(3.13)
Theorem 3.4. Let the hypotheses of Theorem 3.3 be satisfied. Then, (1.1;-1) has no solutions
of type B1 if either one of the following conditions holds:
(D1 ) condition (2.17) holds, and

n
−1
n
−1
2
 q( j) f
a1/α
limsup
2 (k)
n→∞ k=g(n)
j =k


g( j)−1
1/α2
1

a1/α
1 (i)
> 1,
(3.14)
> 0.
(3.15)
i=n0 ≥0
(D2 ) condition (2.19) holds, and
limsup
n→∞
n
−1
k=g(n)

n
−1
2

a1/α
2 (k)
j =k

q( j) f 
g( j)−1
i =n 0 ≥0
1/α2
1

a1/α
1 (i)
358
On the oscillation of certain third-order difference equations
Theorem 3.5. Let conditions (i)–(iv) and (2.3) hold, g(n) = n − τ, n ≥ n0 ∈ N where τ is a
positive integer, and assume that there exists an integer τ > 0 such that τ > τ. If the first-order
delay equation

∆y(n) + q(n) f 
n−
τ −1
 
1/α1 
n
−τ
1
2
 
 
a1/α
a1/α
 f y 1/(α1 α2 ) [n − τ] = 0
1 ( j) f 
2 ( j)
j =n 0
j =n −τ
(3.16)
is oscillatory, then (1.1;-1) has no solution of type B1 .
Next, we will present some results for the nonexistence of solutions of type B3 for
(1.1;-1).
Theorem 3.6. Let conditions (i)–(iv) and (2.3) hold, g(n) > n + 1 for n ≥ n0 ∈ N, and
assume that there exists a nondecreasing sequence {η(n)} such that g(n) > η(n) > n + 1 for
n ≥ n0 . Then, (1.1;-1) has no solution of type B3 if either one of the following conditions
holds:
(E1 ) condition (2.8) holds, and
η(n)−1
limsup
n→∞

q(k) f 
g(k)−1
k =n
 
1/α1 
η(k)−1
1
2
 
 
a1/α
a1/α
 > 1,
1 ( j) f 
2 ( j)
(3.17)
j =η(n)
j =η(k)
(E2 )
f
η(n)−1
limsup
n→∞

q(k) f 
k =n
u
g(k)−1
−→ 0
as u −→ ∞,
(3.18)
 
1/α1 
η(k)−1
1
2
 
 
a1/α
a1/α
 > 0.
1 ( j) f 
2 ( j)
(3.19)
u1/(α1 α2 )
j =η(n)
j =η(k)
Proof. Let {x(n)} be a solution of (1.1;-1) of type B3 . Then there exists a large integer
n0 ∈ N such that (3.2) holds for n ≥ n0 . Now
x(σ) = x(τ) +

≥
σ
−1
j =τ
σ
−1
∆x( j) = x(τ) +
σ
−1
1/α1
1
a1/α
1 ( j)L1 x( j)
j =τ

1
 1/α1
a1/α
1 ( j) L1 x(τ)
(3.20)
for σ ≥ τ ≥ n0 .
j =τ
Letting σ = g(n), τ = η(n) in (3.20), we see that

g(n)−1
x g(n) ≥ 
j =η(n)

1
 1/α1
a1/α
1 ( j) L1 x η(n)
for n ≥ n1 ≥ n0 .
(3.21)
Ravi P. Agarwal et al. 359
Using (3.21) in (1.1;-1) and letting y(n) = L1 x(n), n ≥ n1 we have

g(n)−1

1/α
α
1 ∆
∆y(n) 2 ≥ q(n) f 
a1 1 ( j) f y 1/α1 η(n)
a2 (n)
j =η(n)
for n ≥ n1 .
(3.22)
Clearly, y(n) > 0 and ∆y(n) > 0 for n ≥ n1 . As in the above proof, we can easily find

y η(k) ≥ 
η(k)−1

2
 L1/α2 y η(n)
a1/α
2 ( j)
for k ≥ n − 1 ≥ n1 ,
(3.23)
j =η(n)
where Ly(n) = (∆y(n))α2 /a2 (n). Using (2.3) and (3.23) in (3.22), we have

∆ Ly(k) ≥ q(k) f 
g(k)−1
 
1/α1 
η(k)−1
1
2
 
 
a1/α
a1/α
 f L1/(α1 α2 ) y η(n)
1 ( j) f 
2 ( j)
j =η(k)
j =η(k)
(3.24)
for k ≥ n − 1 ≥ n1 . Summing (3.24) from n to η(n) − 1 ≥ n, we have
Ly η(n) ≥ Ly η(n) − Ly(n)
η(k)−1
≥

q(k) f 
g(k)−1
k =n
 
1/α1 
g(k)−1

1
2


 
a1/α
a1/α
 f L1/(α1 α2 ) y η(k) ,
1 ( j) f 
2 ( j)
j =η(n)
j =η(k)
(3.25)
or
 

1/α1 

.
η(k)−1
g(k)−1
g(k)−1
1/α
1/α
Ly η(k)
1


f
≥
q(k)
f
a
(
j)
a2 2 ( j)

1
1/(α
α
)
1
2
f L
y η(n)
j =η(n)
j =η(n)
k =n
(3.26)
Taking limsup of both sides of (3.26) as n → ∞ and applying the hypotheses, we arrive at
the desired contradiction.
Theorem 3.7. Let the hypotheses of Theorem 3.6 be satisfied. Then, (1.1;-1) has no solution
of type B3 if either one of the following conditions holds:
(F1 ) condition (2.17) holds, and
η(n)−1
limsup
n→∞
k =n

k
−1
2

a1/α
2 (k)
j =n

q( j) f 
g( j)−1
i=η( j)
1/α2
1

a1/α
1 (i)
> 1,
(3.27)
360
On the oscillation of certain third-order difference equations
(F2 )
u
−→ 0 as u −→ ∞,
f 1/α2 u1/α1
η(n)−1
limsup
n→∞

2

a1/α
2 (k)
k
−1

q( j) f 
g( j)−1
j =n
k =n
(3.28)
1/α2
1

a1/α
1 (i)
> 0.
(3.29)
i=η( j)
Proof. Let {x(n)} be a solution of (1.1;-1) of type B3 . As in the proof of
Theorem 3.6, we obtain the inequality (3.22) and we see that y(n) > 0 and ∆y(n) > 0
for n ≥ n1 . Summing inequality (3.22) from n to k − 1 ≥ n ≥ n2 ≥ n1 , we have


g( j)−1
1/α
α
1 ∆y(k) 2 ≥ q( j) f 
a1 1 (i) f y 1/α1 η( j)
a2 (k)
j =n
i=η( j)
k −1
(3.30)
which implies that

k
−1
2

(k)
q( j) f
∆y(k) ≥ a1/α
2
j =n


g( j)−1

1

a1/α
1 (i)
f y
1/α1
1/α2
η( j) 
for n ≥ n2 .
(3.31)
i=η( j)
Combining (3.31) with the relation
y(s) = y(n) +
s
−1
∆y(k) for s − 1 ≥ n ≥ n2
(3.32)
k =n
and setting s = η(n), we have


1/α2
η(n)−1
g( j)−1
k
−1
1/α
1/α
y η(n)
2


≥
a
(k)
q(
j)
f
a1 1 (i)
2
f 1/α2 u1/α1 η(n)
j =n
i=η( j)
k =n
for n ≥ n2 .
(3.33)
Taking limsup of both sides of (3.33) as n → ∞, we arrive at the desired contradiction.
Theorem 3.8. Let conditions (i)–(iv) and (3.2) hold, g(n) = n + σ for n ≥ n0 ∈ N, where σ
is a positive integer, and assume that there exist two positive integers σ and σ̃ > 1 such that
σ − 2 > σ − 1 > σ̃. If the first-order advanced equation

∆y(n) − q(n) f 
n+σ
−1
j =n+σ
 
1/α1 
n+σ −1
 1/α2   1/(α1 α2 )
f
a2 ( j)
[n + σ̃] = 0
f y
1

a1/α
1 ( j)
j =n+σ̃
(3.34)
is oscillatory, then (1.1;-1) has no solution of type B3 .
Ravi P. Agarwal et al. 361
Proof. Let {x(n)} be a solution of (1.1;-1) of type B3 . As in the proof of Theorem 3.6, we
obtain the inequality (3.21) for n ≥ n1 , that is,

x[n + σ] ≥ 
n+σ
−1

1
 1/α1
a1/α
1 ( j) L1 x[n + σ]
for n ≥ n1 .
(3.35)
for n ≥ n2 ≥ n1 .
(3.36)
j =n+σ
Similarly, we see that


n+σ
−1
L1 x[n + σ] ≥ 
2
2
 L1/α
a1/α
2 ( j)
2 x[n + σ̃]
j =n+σ̃
Combining (3.35) and (3.36), we have

x[n + σ] ≥ 
n+σ
−1

1

a1/α
1 ( j)
j =n+σ
n+σ
−1
1/α1
1/α2
1 α2 )
a2 ( j) L1/(α
x[n + σ̃]
1
for n ≥ n2 .
(3.37)
j =n+σ̃
Using (2.3) and (3.37) in (1.1;-1) and letting Z(n) = L1 x(n), n ≥ n2 , we have

n+σ
−1
∆Z(n) ≥ q(n) f 
 
1/α1 
n+σ
−1
1
2
 
 
a1/α
a1/α
 f Z 1/(α1 α2 ) [n + σ̃] .
1 ( j) f 
2 ( j)
j =n+σ
(3.38)
j =n+σ̃
By a known result in [2, 12], we see that (3.34) has an eventually positive solution, a
contradiction. This completes the proof.
Next, we will combine our earlier results to obtain some sufficient conditions for the
oscillation of (1.1;-1), as an example, we state the following result.
Theorem 3.9. Let conditions (i)–(iv) and (2.3) hold, g(n) = n + σ for n ≥ n0 ∈ N, and
assume that there exist two positive integers σ, σ̃ such that σ − 2 > σ − 1 > σ̃. If condition
(3.3) holds and equation (3.34) is oscillatory, then (1.1;-1) is oscillatory.
Proof. The proof follows from Theorems 3.1 and 3.8.
Now, we apply Theorem 3.9 to a special case of (1.1;-1), namely, the equation
α
1 1
∆
∆
∆x(n) 1
a2 (n) a1 (n)
α2 − q(n)xα [n + σ] = 0,
(3.39)
where α is the ratio of positive odd integers and σ is a positive integer, and obtain the
following immediate result.
Corollary 3.10. Let conditions (i)–(iv) hold and assume that there exist two positive integers σ and σ̃ > 1 such that σ − 2 > σ − 1 > σ̃. Then, (3.39) is oscillatory if either one of the
following conditions is satisfied:
362
On the oscillation of certain third-order difference equations
(J1 ) condition (3.3) holds, and
liminf
n→∞
n+σ̃
−1

α 
q(k) 
1
 
a1/α
1 ( j)
k=n+1
k+σ
−1
j =k+σ
k+σ
−1
α2
2
 >
a1/α
2 ( j)
j =k+σ̃
σ̃ − 1
σ̃
σ̃
if α = α1 α2 ,
(3.40)
(J2 ) condition (3.3) holds, and
α 
α/α1
k+σ
−1 1/α
1/α
limsup
q(k) 
a1 1 ( j) 
a2 2 ( j)
n→∞ k=n+1
j =k+σ
j =k+σ̃
n+σ̃
−1

k+σ
−1
>0
if α > α1 α2 .
(3.41)
Now we will combine Theorems 3.5 and 3.8 to obtain some interesting oscillation
criteria for the mixed type of equations
L3 x(n) − q1 (n) f1 x g1 (n) − q2 (n) f2 x g2 (n)
= 0,
(3.42)
where L3 , qi , gi , and fi , i = 1,2 are as in (2.56).
Theorem 3.11. Let the sequences {qi (n)}, {gi (n)}, and fi (x), i = 1,2 be as in (2.56), let L3
be defined as in (1.1;δ), and {ai (n)}, αi , i = 1,2 are as in (i) and (iv), g1 (n) = n − τ and
g2 (n) = n + σ, n ≥ n0 ∈ N, where τ and σ are positive integers. Moreover, assume that there
exist positive integers τ, σ, and σ̃ such that τ > τ and σ − 2 > σ − 1 > σ̃. If (3.16) with q
and f replaced by q1 and f1 , respectively, and (3.34) with q and f replaced by q2 and f2 ,
respectively, are oscillatory, then (3.42) is oscillatory.
Remark 3.12. The results of this paper are presented in a form which is essentially new
even if α1 = α2 = 1.
4. Applications
We can apply our results to neutral equations of the form
L3 x(n) + p(n)x τ(n) + δ f x g(n)
= 0,
(4.1;δ)
where { p(n)} and {τ(n)} are real sequences, τ(n) is increasing, τ −1 (n) exists, and
limn→∞ τ(n) = ∞. Here, we set
y(n) = x(n) + p(n)x τ(n) .
(4.2)
If x(n) > 0 and p(n) ≥ 0 for n ≥ n0 ≥ 0, then y(n) > 0 for n ≥ n1 ≥ n0 . We let 0 ≤ p(n) ≤ 1,
p(n) ≡ 1 for n ≥ n0 , and consider either (P1 ) τ(n) < n when ∆y(n) > 0 for n ≥ n1 , or (P2 )
τ(n) > n when ∆y(n) < 0 for n ≥ n1 . In both cases we see that
x(n) = y(n) − p(n)x τ(n) = y(n) − p(n) y τ(n) − p τ(n) x τ ◦ τ(n)
≥ y(n) − p(n)y τ(n) ≥ y(n) 1 − p(n)
for n ≥ n1 .
(4.3)
Ravi P. Agarwal et al. 363
Next, we let p(n) ≥ 1, p(n) ≡ 1 for n ≥ n0 and consider either (P3 ) τ(n) > n if ∆y(n) > 0
for n ≥ n1 , or (P4 ) τ(n) < n if ∆y(n) < 0 for n ≥ n1 . In both cases we see that
1
x(n) = −1 y τ −1 (n) − x τ −1 (n)
p τ (n)
y τ −1 (n)
y τ −1 ◦ τ −1 (n)
1
−
−
=
p τ −1 (n)
p τ −1 (n) p τ −1 ◦ τ −1 (n)
≥
p
1
τ −1 (n)
1− p
τ −1
x τ −1 ◦ τ −1 (n)
p τ −1 ◦ τ −1 (n)
(4.4)
1
y τ −1 (n)
for n ≥ n1 .
−
1
◦ τ (n)
Using (4.3) or (4.4) in (4.1;δ), we see that the resulting inequalities are of type (1.1;δ).
Therefore, we can apply our earlier results to obtain oscillation criteria for (4.1;δ). The
formulation of such results are left to the reader.
In the case when p(n) < 0 for n ≥ n0 , we let p1 (n) = − p(n) and so
y(n) = x(n) − p1 (n)x τ(n) .
(4.5)
Here, we may have y(n) > 0, or y(n) < 0 for n ≥ n1 ≥ n0 . If y(n) > 0 for n ≥ n0 , we see
that
x(n) ≥ y(n) for n ≥ n1 .
(4.6)
On the other hand, if y(n) < 0 for n ≥ n1 , we have
x τ(n) =
or
y(n)
1 y(n) + x(n) ≥
,
p1 (n)
p1 (n)
(4.7)
y τ −1 (n)
x(n) ≥ −1 p1 τ (n)
for n ≥ n2 ≥ n1 .
(4.8)
Next, using (4.6) or (4.8) in (4.1;δ), we see that the resulting inequalities are of the type
(1.1;δ). Therefore, by applying our earlier results, we obtain oscillation results for (4.1;δ).
The formulation of such results are left to the reader.
Next, we will present some oscillation results for all bounded solutions of (4.1;1) when
p(n) < 0 and τ(n) = n − σ, n ≥ n0 and σ is a positive integer.
Theorem 4.1. Let τ(n) = n − σ, σ is a positive integer, p1 (n) = − p(n) and 0 < p1 (n) ≤ p <
1, n ≥ n0 , p is a constant, and g(n) < n for n ≥ n0 . If
u
≤1
f 1/(α1 α2 ) (u)
limsup
n→∞
n
−1
k=g(n)


a1 (k)
n
−1
j =k
for u = 0,

a2 ( j)
∞
i= j
then all bounded solutions of (4.1;1) are oscillatory.
1/α2 1/α1

q(i)  > 1,
(4.9)
(4.10)
364
On the oscillation of certain third-order difference equations
Proof. Let {x(n)} be a bounded nonoscillatory solution of (4.1;1), say, x(n) > 0 for n ≥
n0 ≥ 0. Set
y(n) = x(n) − p1 (n)x[n − σ] for n ≥ n1 ≥ n0 .
(4.11)
Then,
L3 y(n) = −q(n) f x g(n)
for n ≥ n1 .
≤0
(4.12)
It is easy to see that y(n), L1 y(n), and L2 y(n) are of one sign for n ≥ n2 ≥ n1 . Now, we
have two cases to consider: (M1 ) y(n) < 0 for n ≥ n2 , and (M2 ) y(n) > 0 for n ≥ n2 .
(M1 ) Let y(n) < 0 for n ≥ n2 . Then either ∆y(n) < 0, or ∆y(n) > 0 for n ≥ n2 . If ∆y(n) <
0 for n ≥ n2 , then
x(n) < px[n − σ] < p2 x[n − 2σ] < · · · < pm x[n − mσ]
(4.13)
for n ≥ n2 + mσ, which implies that limn→∞ x(n) = 0. Consequently, limn→∞ y(n) = 0, a
contradiction.
Now, we have y(n) < 0 and ∆y(n) > 0 for n ≥ n2 . Set Z(n) = − y(n) for n ≥ n2 . Then,
L3 Z(n) = q(n) f x g(n)
for n ≥ n2
≥0
(4.14)
and ∆Z(n) < 0 for n ≥ n2 . It is easy to derive at a contradiction if either L2 Z(n) > 0 or
L2 Z(n) < 0 for n ≥ n2 . The details are left to the reader.
(M2 ) Let y(n) > 0 for n ≥ n2 . Then, x(n) ≥ y(n) for n ≥ n2 and from (4.12), we have
L3 y(n) ≤ −q(n) f y g(n)
for n ≥ n2 .
(4.15)
We claim that ∆y(n) < 0 for n ≥ n2 . Otherwise, ∆y(n) > 0 for n ≥ n2 and hence we see that
y(n) → ∞ as n → ∞, a contradiction. Thus, we have y(n) > 0 and ∆y(n) < 0 for n ≥ n2 .
Summing (4.15) from n ≥ n2 to u and letting u → ∞, we have

∞
1/α2
α
1 ∆y(n) 1 ≥ f 1/α2 y g(n) a2 (n) q(i)
∆
a1 (n)
i=n
.
(4.16)
Ravi P. Agarwal et al. 365
Again summing (4.16) twice from j = k to n − 1, and from k = g(n) to n − 1, we obtain
1≥
n
−1

n
−1

∞
y g(n)

a2 ( j)
≥
a1 (k)
f 1/(α1 α2 ) y g(n)
i= j
k=g(n)
j =k
1/α2 1/α1

q(i)  .
(4.17)
Taking limsup of both sides of the above inequality as n → ∞, we arrive at the desired
contradiction. This completes the proof.
In the case when p(n) ≡ −1, we have the following result.
Theorem 4.2. Let τ(n) = n − σ, σ is a positive integer, p(n) = −1, and g(n) < n for n ≥ n2 .
If


1/α2 1/α1
∞
∞
∞

a2 ( j) q(i) 
= ∞,

a1 (k)
(4.18)
i= j
j =k
then all bounded solutions of (4.1;1) are oscillatory.
Proof. Let {x(n)} be a nonoscillatory solution of (4.1;1), say, x(n) > 0 for n ≥ n0 ≥ 0. Set
y(n) = x(n) − x[n − σ] for n ≥ n1 ≥ n0 .
(4.19)
Then,
L3 y(n) = −q(n) f x g(n)
≤0
for n ≥ n1 .
(4.20)
It is easy to check that there are two possibilities to consider: (Z1 ) L2 y(n) ≥ 0, ∆y(n) ≤ 0,
and y(n) < 0 for n ≥ n2 ≥ n1 , or (Z2 ) L2 y(n) ≥ 0, ∆y(n) ≤ 0, and y(n) > 0 for n ≥ n2 .
In case (Z1 ), there exists a finite constant b > 0 such that limn→∞ y(n) = −b. Thus,
there exists an n3 ≥ n2 such that
−b < y(n) < −
b
2
for n ≥ n3 .
(4.21)
Hence,
b
2
for n ≥ n3 ,
(4.22)
b
2
for n ≥ n4 .
(4.23)
b
q(n) for n ≥ n4 .
2
(4.24)
x[n − σ] >
then there exists an n4 ≥ n3 such that
x g(n) >
From (4.20), we have
L3 y(n) ≤ − f
366
On the oscillation of certain third-order difference equations
In case (Z2 ), we have
x(n) ≥ x[n − τ] for n ≥ n2 .
(4.25)
Then there exist a constant b1 > 0 and an integer n3 ≥ n2 such that
for n ≥ n3 .
x g(n) ≥ b1
(4.26)
Hence,
L3 y(n) ≤ − f (b1 )q(n) for n ≥ n4 ≥ n3 .
(4.27)
In both cases we are lead to the same inequality (4.27). Summing (4.27) from n ≥ n4 to
u ≥ n and letting u → ∞, we get

∞
1/α2
α
1 ∆y(n) 1 ≥ f 1/α2 (b1 ) a2 (n) q(i)
∆
a1 (n)
i=n
.
(4.28)
Once again, summing the above inequality from n ≥ n4 to T ≥ n and letting T → ∞, we
have


1/α2 1/α1
∞
∞

a2 (k) q(i) 
−∆y(n) ≥ f 1/(α1 α2 ) (b) a1 (n)
.

n =k
(4.29)
i=k
Summing the above inequality from n4 to n − 1 ≥ n4 , we get
∞ > y(n4 ) > − y(n) + y(n4 ) ≥ f 1/(α1 α2 ) (b1 )
n
−1


1/α2 1/α1
∞
∞

a2 ( j) q(i) 
a1 (k)

k =n 4
−→ ∞
j =k
j =i
as n −→ ∞,
which is a contradiction. This completes the proof.
(4.30)
Acknowledgment
The authors are grateful to Professors M. Migda and Z. Dosla for their comments on the
first draft of this paper.
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Ravi P. Agarwal: Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA
E-mail address: agarwal@fit.edu
Said R. Grace: Department of Engineering Mathematics, Faculty of Engineering, Cairo University,
Orman, Giza 12221, Egypt
E-mail address: srgrace@eng.cu.edu.eg
Donal O’Regan: Department of Mathematics, National University of Ireland, Galway, University
Road, Galway, Ireland
E-mail address: donal.oregan@nuigalway.ie